Theory of Diffraction- gratings. 195 



and the resultant will therefore be nil. The same result must 

 ensue whenever B D is an exact multiple of A. 



For the intermediate directions we require a little more cal- 

 culation. 



The phase of the resultant will always correspond with that of 

 the secondary wave which issues from the middle of the aperture. 

 If x denotes the retardation of any element with respect to this 

 one, the amplitude of the resultant is given by 



■+* 



1 T^ o 



I R cos x da?-T-H f 



J~2 



where R is the relative retardation of the extreme parts A and B, 

 or, on integration, 



. R . R 



This expression gives the magnitude of the resultant amplitude 

 compared with that in the principal direction B C, where all the 

 components agree in phase. 



The composition of elementary vibrations whose phases vary 

 uniformly within certain limits may be illustrated by a mecha- 

 nical analogy. Each elementary vibration is represented by a 

 force proportional to the element of circular arc P Q and acting 

 at along a direction P, making with a fixed line of reference 

 X an angle corresponding to the 

 phase of the vibration. The force 

 may be supposed to be due to the 

 attraction of the arc on a particle 

 placed at O. The group of vibra- 

 tions is thus represented by the 

 group of forces whose directions are 

 distributed uniformly through the 

 angle A B ; and the resultant of 

 the forces, found by resolving in 

 the ordinary way, represents on the 

 same system the resultant of the 

 vibrations. In the present case A B corresponds to R, and 



the integrated expression sin — -r- — denotes the ratio of the re- 



sultant force to the aggregate of its components calculated with- 

 out allowance for the difference of direction — that is, as if the 

 whole attracting mass were concentrated at X. 



xiccording to what has been already explained, ( sin-^ -r ) 



2 



